Identifying reservoir depletion patterns from production-induced deformations with applications to seismic imaging

Appendix A

The elastostatic Green's tensor $g^k_i(x,y,z,\xi,\eta,\zeta)$ has the meaning of the displacement along axis $i$ at point $(x,y,z)$ due to a concentrated force along axis $k$ at point $(\xi,\eta,\zeta)$ . The analytical expression for the components of the Green's tensor in the elastic half-space with a free-surface boundary condition are given by the following equations (Mindlin, 1936):

$$g^1_1= w \left( \frac{3-4\nu}{r_1}+\frac{1}{r_2}+ \frac{(x-\xi)}{r^3_1}+\frac{(3-4\nu)(x-\xi)^2}{r^3_2}\right)+$$

$$+ w\left(\frac{2(r^2_2-3(x-\xi)^2)z\zeta}{r^5_2}+\frac{4(1-\nu)(1-2\nu)(r^2_2-(x-\xi)^2-r_2(z+\zeta))}{r_2(r_2-z-\zeta)^2} \right)$$

$$g^1_2= (x-\xi)(y-\eta)w\left(\frac{1}{r^3_1}+ \frac{3-4\nu}{r^3_2}- \frac{6z\zeta}{r^5_2}-\frac{4(1-\nu)(1-2\nu)}{r_2(r_2-z-\zeta)^2} \right)$$

$$g^1_3= (x-\xi)w\left( \frac{z-\zeta}{r^3_1}+\frac{(3-4\nu)(z-\zeta)}{r^3... ...{4(1-\nu)(1-2\nu)}{r_2(r_2-z-\zeta)}- \frac{6 z \zeta (z+\zeta)}{r^5_2} \right)$$

$$g^2_1= g^1_2$$

$$g^2_2= w \left( \frac{3-4\nu}{r_1}+\frac{1}{r_2}+ \frac{(y-\eta)}{r^3_1}+\frac{(3-4\nu)(y-\eta)^2}{r^3_2}\right)+$$

$$+ w\left(\frac{2(r^2_2-3(y-\eta)^2)z\zeta}{r^5_2}+\frac{4(1-\nu)(1-2\nu)(r^2_2-(y-\eta)^2-r_2(z+\zeta))}{r_2(r_2-z-\zeta)^2} \right)$$

$$g^2_3= (y-\eta)w\left( \frac{z-\zeta}{r^3_1}+ \frac{(3-4\nu)(z-\zeta)}{r... ...{4(1-\nu)(1-2\nu)}{r_2(r_2-z-\zeta)}- \frac{6 z \zeta (z+\zeta)}{r^5_2} \right)$$

$$g^3_1= (x-\xi)w \left( \frac{z-\zeta}{r^3_1}+ \frac{(3-4\nu)(z-\zeta)}{r... ...rac{4(1-\nu)(1-2\nu)}{r_2(r_2-z-\zeta)}+ \frac{6z\zeta(z+\zeta)}{r^5_2} \right)$$

$$g^3_2= (y-\eta) w \left( \frac{z-\zeta}{r^3_1}+ \frac{(3-4\nu)(z-\zeta)}... ...rac{4(1-\nu)(1-2\nu)}{r_2(r_2-z-\zeta)}+ \frac{6z\zeta(z+\zeta)}{r^5_2} \right)$$

$$g^3_3= w\left( \frac{3-4 \nu}{r_1}+\frac{5-12\nu+8\nu^2}{r_2}+ \frac{(z-... ...-4\nu)(z+\zeta)^2-2z \zeta}{r^3_2}+ \frac{6 z \zeta (z+\zeta)^2}{r^5_2} \right)$$

$$w= \frac{1}{16 \pi \mu (1-\nu)}$$

$$r_1= \sqrt{(x-\xi)^2+(y-\eta)^2+(z-\zeta)^2}$$

$$r_2= \sqrt{(x-\xi)^2+(y-\eta)^2+(z+\zeta)^2}$$

Identifying reservoir depletion patterns from production-induced deformations with applications to seismic imaging